Find the absolute extrema of the function on the closed interval.
y = 3 cos x, [0, 2π]

The present value of 350,000 due after 10 years, compounded annually at an interest rate of 8% per year, is approximately 157,456.41, while the present value of 350,000 due after 10 years, compounded continuously at an interest rate of 8% per year, is approximately 147,252.13.

i. To find the present value of 350,000 due after 10 years, compounded annually at an interest rate of 8% per year, we can use the formula:

[tex]PV = FV / (1 + r)^n[/tex]

where PV is the present value, FV is the future value, r is the interest rate, and n is the number of periods.

Substituting the given values, we get:

PV = 350,000 / (1 + 0.08)^10

PV ≈ 157,456.41

Therefore, the present value of 350,000 due after 10 years, compounded annually at an interest rate of 8% per year, is approximately 157,456.41.

ii. To find the present value of 350,000 due after 10 years, compounded continuously at an interest rate of 8% per year, we can use the formula:

[tex]PV = FV / e^(r * n)[/tex]

where e is the mathematical constant approximately equal to 2.71828.

Substituting the given values, we get:

[tex]PV = 350,000 / e^(0.08 * 10)[/tex]

PV ≈ 147,252.13

Therefore, the present value of 350,000 due after 10 years, compounded continuously at an interest rate of 8% per year, is approximately 147,252.13.

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Let u = tan⁻¹x, then du/dx = 1/(1+x²).

Using the formula for the derivative of inverse tangent, we have:

tan(u) = x

sec²(u) du/dx = 1

du/dx = cos²(u)

Substituting into the original expression, we get:

∫(tan⁻¹x × e^tan⁻¹x)/(1+x²) dx = ∫(tan⁻¹x × e^u × cos²(u)) du

Using integration by parts with u = tan⁻¹x and dv = e^u × cos²(u) du, we get:

v = (1/2) e^u (sin(u) + u cos(u))

∫(tan⁻¹x × e^u × cos²(u)) du = (1/2) e^u (sin(u) + u cos(u)) tan⁻¹x - ∫[(sin(u) + u cos(u)) / (1+x²)] dx

Substituting back u = tan⁻¹x, we get:

∫(tan⁻¹x × e^tan⁻¹x × cos²(tan⁻¹x)) dx = (1/2) e^tan⁻¹x (x sin(tan⁻¹x) + cos(tan⁻¹x)) tan⁻¹x - ∫[(x cos(tan⁻¹x) + sin(tan⁻¹x)) / (1+x²)] dx

Using the identity sin(tan⁻¹x) = x / √(1+x²) and cos(tan⁻¹x) = 1 / √(1+x²), we simplify the expression to:

∫(tan⁻¹x × e^tan⁻¹x × cos²(tan⁻¹x)) dx = (1/2) x e^tan⁻¹x + (1/2) ∫[e^tan⁻¹x / (1+x²)] dx

The remaining integral can be solved using another substitution with v = tan⁻¹x, which results in:

∫(tan⁻¹x × e^tan⁻¹x × cos²(tan⁻¹x)) dx = (1/2) x e^tan⁻¹x + (1/2) ln(1+x²) + C, where C is the constant of integration

Answer:

Step-by-step explanation:

SOHCAHTOA    [tex]sin =\frac{opp}{hyp} ,cos=\frac{adj}{hyp} ,tan=\frac{opp}{adj}[/tex]

[tex]sin60=\frac{x}{8}[/tex]

[tex]x=8sin60=8\frac{\sqrt{3} }{2} =4\sqrt{3}[/tex]

Answer: 44

Step-by-step explanation: So, if the area of a square is 121 sq. in, the sides that could be multiplied to get 121 are 11 x 11. Then, you multiply by four to get the perimeter.

f(x) = (3/2)*x² + (13/2)*x - 5/2 - 6/(x + 3) + 15/(x - 5)

and the values of P = 3/2, Q = 13/2, R = -5/2, V = -6, W = 15

What is the fraction?

A fraction is a number that represents a part of a whole or a ratio between two quantities. It is written with a numerator (the top number) and a denominator (the bottom number) separated by a horizontal line. The numerator represents the number of parts being considered, while the denominator represents the total number of parts that make up a whole unit.

First, let's factor in the denominator of the given function:

x² - 2x - 15 = (x - 5)(x + 3)

Now we can use partial fraction decomposition to express the given function as the sum of simpler fractions:

f(x) = (x⁴ + 2x³ - 29x² - 47x + 77) / (x² - 2x - 15)

= (x⁴ + 2x³ - 29x² - 47x + 77) / ((x - 5)(x + 3))

= (Px² + Qx + R) + V/(x + 3) + W/(x - 5)

where P, Q, and R are constants to be determined, and V and W are constants to be determined as well.

To find the values of P, Q, and R, we can use polynomial long division to divide the numerator by the denominator:

        x² + 5x - 2

x² - 2x - 15 | x⁴ + 2x³ - 29x² - 47x + 77

- (x⁴ - 2x³ - 15x²)

----------------------

4x³ - 14x² - 47x

- (4x³ - 8x² - 60x)

------------------

-6x² + 47x

- (-6x² + 12x + 90)

--------------

35x - 90

Therefore,

f(x) = (x⁴ + 2x³ - 29x² - 47x + 77) / (x² - 2x - 15)

= x² + 5x - 2 - (6x² - 35x + 90) / ((x - 5)(x + 3))

= -5/2 + (13/2)x + (3/2)x² + V/(x + 3) + W/(x - 5)

Comparing this to the given form, we can see that:

P = 3/2

Q = 13/2

R = -5/2

V = -6

W = 15

Therefore, we have:

f(x) = (3/2)*x² + (13/2)*x - 5/2 - 6/(x + 3) + 15/(x - 5)

and the values of P = 3/2, Q = 13/2, R = -5/2, V = -6, W = 15

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The width and length of the rectangle will be -1 units and 3 units, respectively.

Let W be the rectangle's width and L its length.

The area of the rectangle is the multiplication of the two different sides of the rectangle. Then the rectangle's area will be

[tex]\text{Area of the rectangle = L x W square units}[/tex]

The width of a square shape is 4 units less than the length. The region of the square shape is 12 square units. Then the equations are given below.

[tex]W = L - 4[/tex]              ...1

[tex]L \times W = 12[/tex]           ...2

From equations 1 and 2, then we have

[tex]L \times (L - 4) = 12[/tex]

[tex]L^2 - 4L = 12[/tex]

[tex]L^2 - 4L - 12 = 0[/tex]

[tex]L^2 - 3L + 4L - 12 = 0[/tex]

[tex]L(L - 3) + 4(L - 3) = 0[/tex]

[tex](L - 3)(L + 4) = 0[/tex]

[tex]L = 3, -4[/tex]

Then the width of the rectangle is given as,

[tex]W = 3 - 4[/tex]

[tex]W = -1 \ \text{units}[/tex]

The width and length of the rectangular shape will be -1 units and 3 units, separately.

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Answer:

Step-by-step explanation:

[tex]P+Q-R=(2x^2+3xy-5y^2)+(-5x^2+2xy+3y^2)-(-3x^2+5xy-2y^2)[/tex]

                  [tex]=2x^2+3xy-5y^2-5x^2+2xy+3y^2+3x^2-5xy+2y^2[/tex]

                  [tex]=2x^2-5x^2+3x^2-5y^2+3y^2+2y^2+3xy+2xy-5xy[/tex]

                  [tex]=0[/tex]

Answer:

To show that P + Q - R = 0, we need to substitute the expressions for P, Q, and R into the equation and simplify.

P + Q - R = (2x² + 3xy - 5y²) + (-5x² + 2xy + 3y²) - (-3x² + 5xy - 2y²)

Simplifying the brackets on the right-hand side:

P + Q - R = 2x² + 3xy - 5y² - 5x² + 2xy + 3y² + 3x² - 5xy + 2y²

Grouping like terms together:

P + Q - R = (2x² - 5x² + 3x²) + (3xy + 2xy - 5xy) + (-5y² + 3y² + 2y²)

Simplifying the expressions in the brackets:

P + Q - R = 0x² + 0xy + 0y²

Therefore, P + Q - R = 0.

There is a missing step between step 8 and 9 because the student did not state that m∠BOC + m∠COD = m∠BOD. This statement is true because of the angles addition postulate.

The Angle Addition Postulate states that if point P lies in the interior of angle ∠RST, then the measure of ∠RST is equal to the sum of the measures of ∠RSP and ∠PST. In other words, for any angle ∠RST with point P in its interior, says that m∠RST = m∠RSP + m∠PST.

The Angle Addition Postulate is used in geometry to find the measures of angles that are formed by two intersecting lines or when a point is located in the interior of an angle.

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Image transcribed:

Select the correct answer from each drop-down menu.

Given: ∠BOC and ∠COD are complementary angles

lines BO intersects lines AD at point O

Prove: ∠AOB ≅∠BOD

Statements--|--Reasons

1. ∠BOC and ∠COD are complementary angles; BO intersects AD at point O--|--1. given

2. m∠BOC+ m∠COD = 90°--|--2. definition of complementary angles

3. substitution property of equality--|--3. m∠BOD 90°

4. ∠AOB and ∠BOD are supplementary angles--|--4. linear pair theorem

5. m∠AOB + m∠BOD = 180°--|--5. definition of supplementary angles

6. m∠AOB+90° = 180°--|--6. substitution property of equality

7. m∠AOB = 90°--|--7. subtraction property of equality

8. m∠AOB = m/BOD--|--8. substitution property of equality

9. ∠AOB∠ BOD--|--9. definition of congruent angles

There is a step missing in the proof. Identify where in the proof there is a missing step and what this step should be.

There is a missing step between _______ (options 8. and 9.,1. and 2., 2 and 3., 4. and 6.)because the student did not state m∠BOC+m∠COD - m∠______  (options AOB or BOD). This statement is true because of the _____ (options angles addition postulate, d definition of complementary angles, addition, property of equality, definition of perpendicular line segments)

Answer:

Step-by-step explanation:

The price that maximizes revenue is $10 per ticket, which will bring in $1,500,000 in revenue. 60,000 people will attend at that price.

Price is the amount of money that is required to purchase or obtain a good or service. It is a numerical value that is assigned to a particular product or service and is used to indicate its value in the market. The price of a product or service is determined by a variety of factors, including the cost of production, supply and demand, competition, and market conditions.

In the given question,

To maximize revenue, we need to find the price per ticket that will bring in the greatest total revenue. Let's start by figuring out how many people will attend at various price points:

At $20 per ticket, 40,000 people attend.

At $15 per ticket, 50,000 people attend.

At $10 per ticket, 60,000 people attend.

At $5 per ticket, 70,000 people attend.

Now we can calculate the total revenue at each price point:

At $20 per ticket, revenue is 40,000 × $20 + 40,000 × $5 = $1,000,000.

At $15 per ticket, revenue is 50,000 × $15 + 50,000 × $5 = $1,250,000.

At $10 per ticket, revenue is 60,000 × $10 + 60,000 × $5 = $1,500,000.

At $5 per ticket, revenue is 70,000 × $5 + 70,000 × $5 = $700,000.

From these calculations, we can see that the price that maximizes revenue is $10 per ticket, which will bring in $1,500,000 in revenue. 60,000 people will attend at that price.

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The answer of the given question based on function is  g(t) = 1.7 * 0.8^t models exponential decay.

Exponential decay is a type of mathematical function that represents a process in which the value of a quantity decreases over time or through a series of events at a constant proportional rate. In exponential decay, the rate of decrease is proportional to the current value of the quantity.

Exponential decay is often modeled by the following equation:

y = a * e^(-kt)

The function g(t) = 1.7 * 0.8^t models exponential decay.

In this function, the base of the exponent is 0.8, which is a number between 0 and 1. When a base of an exponential function is between 0 and 1, it represents exponential decay. This means that as the value of t increases, the value of g(t) decreases at an increasingly rapid rate.

Additionally, the coefficient of the exponent (1.7 in this case) represents the initial value of the function. Since the coefficient is positive, the initial value is also positive. Therefore, the function represents exponential decay from a positive initial value.

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Answer:

Step-by-step explanation:

1+8×6091+689+8×879+68×9+7​

We need to do the multiplications first

1+(8×6091)+689+(8×879)+(68×9)+7​=

1+(48728)+689+(7032)+(612)+7​

Now we add all of them up

48729 + 7721 + 619 =

57069

Answer is 57069

Answer:

8 mi per in.

Step-by-step explanation:

Divide 24 mi by 3 in. to get base number of 1 in. per x mi

24/3 = 8

8 mi per in.

Answer:

-----------------------------------

The scale factor k affects the points with the rule:

Apply the rule to the given points, given k = 1/2:

By pythagorean theorem, 21.095 is the length of the segment indicated..

What is the Pythagorean theorem in plain English?

The Pythagorean Theorem states that the squares on the hypotenuse of a right triangle, which is the side that faces the right angle, are equal when added together. This is written as a2 + b2 = c2 in the usual algebraic notation.

  In triangle we apply pythagorean theorem

    x² = 11.8² + 17.5²

         = 139.24 +  306.25

     x   = √445.49

      x = 21.095

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Step-by-step explanation:

To find the inverse of an equation, we need to switch the roles of the dependent variable and the independent variable. In other words, if we have an equation of the form y = f(x), we need to rewrite it as x = f^{-1}(y), where f^{-1}(y) is the inverse function of f.

Once we have found the inverse equation, we can perform the original steps of the inverse equation by plugging in the output of the inverse function into the original equation.

For example, let's say we have the equation y = 2x + 3. To find the inverse equation, we first switch the roles of x and y to get:

x = 2y + 3

Next, we solve for y in terms of x:

x - 3 = 2y

(y = x - 3)/2

So the inverse equation is y = (x - 3)/2.

To perform the original steps of the inverse equation, we can plug the output of the inverse function, (x - 3)/2, into the original equation, y = 2x + 3:

y = 2((x - 3)/2) + 3

y = x - 3 + 3

y = x

We have arrived back at the independent variable, x, so the inverse and original steps have canceled each other out, as expected.

Answer:

g(0)= -1/3

g(-1)=0

g(1)=-1

g(2)=-3

g(3)=∞

Step-by-step explanation:

g(0)= -1/3

g(-1)=0

g(1)=-1

g(2)=-3

g(3)=∞

Answer:

Yes, (x - 4) is a factor of polynomial function P(x) = x⁵ + 4x⁴ - 17x³ - 60x².

Step-by-step explanation:

The Factor Theorem states that if f(x) is a polynomial, and f(a) = 0, then (x - a) is a factor of f(x).

Therefore, according to the Factor Theorem, if (x - 4) is a factor of P(x) then P(4) = 0.

To determine if (x - 4) is a factor of P(x), substitute x = 4 into the function and solve.

[tex]\begin{aligned}x=4 \implies P(4)&=(4)^5+4(4)^4-17(4)^3-60(4)^2\\&=1024+4(256)-17(64)-60(16)\\&=1024+1024-1088-960\\&=2048-1088-960\\&=960-960\\&=0\end{aligned}[/tex]

As P(4) = 0, this confirms that (x - 4) is a factor of polynomial function P(x) = x⁵ + 4x⁴ - 17x³ - 60x².

[tex]\hrulefill[/tex]

Additional comments

The fully factored form of the given polynomial function is:

[tex]P(x)=x^2\left(x+3\right)\left(x-4\right)\left(x+5\right)[/tex]

The function based on the given parent function and transformations in the given order would be:

f(x)= -1/2√(x+2)

Functions are often represented as equations, graphs, or tables, and they are used to model a wide range of real-world phenomena. For example, a function might describe the relationship between the time of day and the temperature outside, or the amount of money earned based on the number of hours worked.

Let's break down how each transformation affects the parent function:

Shifting units to the left: This transformation involves replacing the variable "x" with "x+2" in the parent function. This has the effect of shifting the graph 2 units to the left. The function becomes:

f(x)= √(x+2)

Stretching horizontally: This transformation involves replacing "x" with "2x" in the function obtained after the first transformation. This has the effect of stretching the graph horizontally by a factor of 1/2. The function becomes:

f(x)= √(2x+4)

Reflecting across the x-axis: This transformation involves multiplying the entire function by -1. This has the effect of reflecting the graph across the x-axis. The function becomes:

f(x)= -√(2x+4)

So the final function based on the given parent function and transformations in the given order is:

f(x)= -1/2√(x+2)

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11 is the value of x in this polygon.

What exactly is a polygon in mathematics?

A closed, two-dimensional, flat, closed polygon is a shape that is constrained by geometry and has straight sides. Its sides don't curve inward at all.

                               Another term for a polygon's sides is its polygonal edges. The points where two sides converge are known as a polygon's vertices (or corners).

2x - 4/6  = 9/3

(2x - 4)3  = 6 * 9

 2x - 4 = 54/3

   2x - 4 = 18

     2x = 18 + 4

       2x = 22

          x = 11

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The numbers which replaces the capital letters are A is -1, B is 1, C is 3, D is 0, E is 5 and F is -2.

A logarithm is a mathematical function that measures the relationship between two quantities, by expressing one quantity in terms of another. More specifically, a logarithm is the power to which a base must be raised to produce a given number.

a) [tex]2^(A) * 2^(B) = 2^0[/tex] can be rewritten as [tex]2^(A+B) = 2^0[/tex].

To find the values of A and B, we need to solve the equation [tex]2^(A+B) = 2^0\\[/tex].

Since [tex]2^0[/tex] = 1, we have [tex]2^(A+B)[/tex] = 1, which means A + B = 0. One possible solution is A = -1 and B = 1, since (-1) + 1 = 0.

Therefore, the equation becomes [tex]2^(-1) * 2^1 = 2^0[/tex], which is true.

(A) = -1

(B) = 1

b) [tex](2^3)/(2^(C)) = 2^(D)[/tex] can be simplified as [tex]2^(3-C) = 2^(D)[/tex].

To find the values of C and D, we need to set the exponents equal to each other. So, we have 3 - C = D, which means C = 3 - D.

One possible solution is C = 3 and D = 0, since 3 - 0 = 3.

Therefore, the equation becomes [tex](2^3)/(2^3) = 2^0[/tex], which is true.

(C) = 3

(D) = 0

c) [tex]2^(-3) * (E)^(-3) = 10^(F)[/tex] can be rewritten as [tex](1/2^3) * (1/E^3) = 10^F[/tex]. Multiplying both sides by [tex]2^3 * E^3[/tex], we get [tex]1 = 10^F * 2^3 * E^3[/tex].

Taking the logarithm of both sides, we get log(1) = [tex]log(10^F * 2^3 * E^3)[/tex]. Since log(1) = 0, we have 0 = F log(10) + 3log(2) + 3log(E), which simplifies to F = -2 - 3log(E)/log(10) with one possible solution of E = 5 and F = -2. Therefore, the equation becomes [tex]2^(-3) * 5^(-3) = 10^(-2)[/tex], which is true.

(E) = 5

(F) = -2

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Therefore, the combined area of the living room and hallway is 731 square feet.

In geometry, the term "area" refers to the measurement of the size of a two-dimensional surface. It is typically measured in square units, such as square meters or square feet. The area of a shape can be found by multiplying its length by its width or by using other specific formulas that depend on the shape itself. Some common examples of shapes for which we might want to calculate the area include squares, rectangles, triangles, circles, and irregular polygons.

by the question.

The area of the living room can be calculated as:

[tex]Area of living room = length x width\\Area of living room = a x b\\Area of living room = 34 ft x 13 ft\\Area of living room = 442 sq ft[/tex]

The area of the hallway can also be calculated using the same formula:

[tex]Area of hallway = length x width\\Area of hallway = c x d\\Area of hallway = 17 ft x 17 ft\\Area of hallway = 289 sq ft[/tex]

To find the total area of the living room and hallway together, we simply add the two areas:

[tex]Total area = Area of living room + Area of hallway\\Total area = 442 sq ft + 289 sq ft\\Total area = 731 sq ft[/tex]

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The area of the archway is (π + 24) square meters, as the total area of the two rectangular pillars is 24 square meters, area of semicircle is (9π) / 2 square meters.

Radius is a term used in geometry to refer to the distance from the center of a circle or sphere to its edge. It is usually represented by the symbol "r". In a circle, all points on the edge, also called the circumference, are at the same distance from the center. This distance is equal to the radius. The radius is half the length of the diameter, which is the distance across the circle passing through its center.

To determine the area of the archway, we need to calculate the area of the two rectangular pillars and the semicircle top arch, and then add them together.

The area of each rectangular pillar is:

Area of each pillar = length x width

= 4 meters x 3 meters

= 12 square meters

So the total area of the two rectangular pillars is:

Total area of pillars = 2 x 12 square meters

= 24 square meters

Now we need to calculate the area of the semicircle top arch. The radius of the semicircle is half of the distance between the two pillars, which is:

Radius of semicircle = (6 meters / 2) = 3 meters

The area of the semicircle is:

Area of semicircle = (π x r²) / 2

= (π x 3²) / 2

= (9π) / 2 square meters

We also need to subtract the area of the rectangle sections at the top of each pillar from the semicircle to get the area of the archway. Each of these rectangles has a length of 3 meters and a height of the difference between the radius of the semicircle and the height of the pillars, which is:

Height of rectangle = (3 meters - 4 meters) = -1 meter

Since we can't have a negative height, we take the absolute value of this difference:

|Height of rectangle| = 1 meter

The area of each rectangle is:

Area of each rectangle = length x width

= 3 meters x 1 meter

= 3 square meters

So the total area of the two rectangles is:

Total area of rectangles = 2 x 3 square meters

= 6 square meters

Therefore, the area of the archway is:

Area of archway = Area of pillars + Area of semicircle - Area of rectangles

= 24 square meters + (9π) / 2 square meters - 6 square meters

= (π + 24) square meters

Hence, the area of the archway is (π + 24) square meters.

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Using the formula for lines intersecting in a circle, we get the value of x to be = 32.5cm.

A diameter is a line segment that traverses a circle by going through its centre.

The radius is twice as long as the diameter.  A chord is a segment that does not have to pass through the centre and has endpoints on the circle.

We know that in a circle when lines are intersecting each other than the lines divided are in the formula:

a × b = c × d

Here,

13 × 20 = 8 × x

⇒ x = 260/8

⇒ x = 32.5cm.

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Answer:

The answer is 3

Step-by-step explanation:

20% of 15 can be written as 20% × 15

= 20/100 × 15

= 3

Thus, 20% of 15 is 3.

hope this helps

if the probability of observing 4 or fewer years with 5 hurricanes is much lower (e.g., less than 5%), then we might conclude that the Poisson distribution does not work well for this particular scenario.

a. The probability of observing 5 hurricanes in a year can be calculated using the Poisson distribution formula:

[tex]P(X=5) = (e^(-λ) * λ^5) / 5![/tex]

where λ is the mean number of hurricanes per year. In this case [tex], λ = 7.7[/tex] . Thus,

[tex]P(X=5) = (e^(-7.7) * 7.7^5) / 5! = 0.0834[/tex]   (rounded to four decimal places)

Therefore, the probability of observing 5 hurricanes in a year is approximately  [tex]0.0834[/tex] or   [tex]8.34[/tex] %.

b. The number of hurricanes in a 45-year period follows a Poisson distribution with mean [tex]λ = 7.7 \times 45 = 346.5[/tex] . The expected number of years with 5 hurricanes in a 45-year period is then:

[tex]E(X) = λ = 346.5[/tex]

Therefore, we expect about [tex]346.5/45 ≈ 7.7[/tex] % or 8 years out of 45 to have 5 hurricanes.

c. If in a recent period of 45 years, only 4 years had 5 hurricanes, we can compare this to the expected number of years with 5 hurricanes based on the Poisson distribution. Using the same mean λ = 346.5, we can calculate the probability of observing 4 or fewer years with 5 hurricanes in a 45-year period:

[tex]P(X ≤ 4) = Σ(i=0 to 4) [(e^(-λ) * λ^i) / i!] ≈ 0.2088[/tex]

This means that there is about a 20.88% chance of observing 4 or fewer years with 5 hurricanes in a 45-year period, based on the Poisson distribution with [tex]λ = 346.5[/tex]  .

Therefore,  This is not a very low probability, so the observed 4 years with 5 hurricanes in the recent [tex]45[/tex]  -year period is not necessarily inconsistent with the Poisson distribution.

Learn more about Poisson here:

https://brainly.com/question/17280826

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Answer-

answer marked on the picture

Step-by-step explanation:

angle 1 and 55 is not vertical angles thus is the wrong answer

angle 2 can be determined with the given info by subtracting the other two angles to 180

Angle 2 not equal to 125 as the angles are not supplementary from each other